| L(s) = 1 | + (−1.63 + 0.675i)2-s + (3.96 + 0.789i)3-s + (−0.624 + 0.624i)4-s + (−4.29 − 6.43i)5-s + (−7.00 + 1.39i)6-s + (−2.27 + 3.40i)7-s + (3.29 − 7.96i)8-s + (6.81 + 2.82i)9-s + (11.3 + 7.58i)10-s + (1.35 + 6.80i)11-s + (−2.97 + 1.98i)12-s + (2.37 + 2.37i)13-s + (1.40 − 7.08i)14-s + (−11.9 − 28.9i)15-s + 11.6i·16-s + (−8.76 + 14.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.815 + 0.337i)2-s + (1.32 + 0.263i)3-s + (−0.156 + 0.156i)4-s + (−0.859 − 1.28i)5-s + (−1.16 + 0.232i)6-s + (−0.324 + 0.486i)7-s + (0.412 − 0.995i)8-s + (0.757 + 0.313i)9-s + (1.13 + 0.758i)10-s + (0.123 + 0.618i)11-s + (−0.247 + 0.165i)12-s + (0.182 + 0.182i)13-s + (0.100 − 0.506i)14-s + (−0.798 − 1.92i)15-s + 0.730i·16-s + (−0.515 + 0.856i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.665538 + 0.145246i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.665538 + 0.145246i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (8.76 - 14.5i)T \) |
| good | 2 | \( 1 + (1.63 - 0.675i)T + (2.82 - 2.82i)T^{2} \) |
| 3 | \( 1 + (-3.96 - 0.789i)T + (8.31 + 3.44i)T^{2} \) |
| 5 | \( 1 + (4.29 + 6.43i)T + (-9.56 + 23.0i)T^{2} \) |
| 7 | \( 1 + (2.27 - 3.40i)T + (-18.7 - 45.2i)T^{2} \) |
| 11 | \( 1 + (-1.35 - 6.80i)T + (-111. + 46.3i)T^{2} \) |
| 13 | \( 1 + (-2.37 - 2.37i)T + 169iT^{2} \) |
| 19 | \( 1 + (-22.7 + 9.43i)T + (255. - 255. i)T^{2} \) |
| 23 | \( 1 + (11.2 - 2.24i)T + (488. - 202. i)T^{2} \) |
| 29 | \( 1 + (-9.98 + 6.67i)T + (321. - 776. i)T^{2} \) |
| 31 | \( 1 + (-7.38 + 37.1i)T + (-887. - 367. i)T^{2} \) |
| 37 | \( 1 + (31.6 + 6.29i)T + (1.26e3 + 523. i)T^{2} \) |
| 41 | \( 1 + (-18.7 + 28.1i)T + (-643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-3.21 - 1.33i)T + (1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-3.16 - 3.16i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (28.3 - 11.7i)T + (1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (4.49 - 10.8i)T + (-2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-58.9 - 39.3i)T + (1.42e3 + 3.43e3i)T^{2} \) |
| 67 | \( 1 + 28.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (33.7 + 6.70i)T + (4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (52.0 + 77.8i)T + (-2.03e3 + 4.92e3i)T^{2} \) |
| 79 | \( 1 + (-4.18 - 21.0i)T + (-5.76e3 + 2.38e3i)T^{2} \) |
| 83 | \( 1 + (-43.4 - 104. i)T + (-4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (28.3 - 28.3i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (138. - 92.4i)T + (3.60e3 - 8.69e3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.09737596074669099542739045119, −17.49106988771116373700500066398, −16.08715468415279784044274348760, −15.35504793912740461673967334970, −13.48204519608541014008585058445, −12.25850728512923143788352400893, −9.507224013604162901297284423054, −8.740907734124751232301824231597, −7.74624421637050607435966712129, −4.08600961077270380115859267313,
3.19692169658130185197820665222, 7.30185644386452081727377037083, 8.518022812927411086323442658732, 10.04004246105681872475083467203, 11.36767615429562852365151586144, 13.80807738048623079257004034538, 14.41872137466452159451442812296, 15.92051701593059035196477507523, 18.02943907391608344835540231383, 18.89920394271429262911958047744
